2.1 Whole Numbers, Place Value, Integers & Order of Operations

Whole Numbers, Place Value, Integers & Order of Operations

Number sense forms the bedrock of the Ontario Mathematics Curriculum (Grades 1–9, Strand B: Number). For elementary and secondary teacher candidates, demonstrating absolute mastery over whole numbers, place value notation, integer arithmetic, and the formal order of operations is essential for passing the Ontario Mathematics Proficiency Test (MPT). This section details the structural rules of arithmetic, highlights common calculation traps, and provides step-by-step worked solutions for complex expressions.

---

The Decimal Place Value System

Our base-10 positional number system assigns value to each digit based on its position relative to the decimal point. Each step to the left increases a digit's place value by a factor of 10 ($10^1$), while each step to the right decreases its value by a factor of 10 ($10^{-1}$ or $\frac{1}{10}$).

Forms of Representation

Numbers can be expressed in three standard mathematical forms:

1. Standard Form: $4,852,739.168$
2. Expanded Form: $(4 \times 10^6) + (8 \times 10^5) + (5 \times 10^4) + (2 \times 10^3) + (7 \times 10^2) + (3 \times 10^1) + (9 \times 10^0) + (1 \times 10^{-1}) + (6 \times 10^{-2}) + (8 \times 10^{-3})$
3. Word Form: Four million, eight hundred fifty-two thousand, seven hundred thirty-nine and one hundred sixty-eight thousandths.

Formal Rounding Algorithm

To round a number to a specified place value:

1. Identify the target digit in the requested place value position.
2. Look at the indicator digit immediately to its right.
3. If the indicator digit is $5$ or greater ($\ge 5$), add $1$ to the target digit and replace all digits to its right with zeros (or drop them if they fall after the decimal point).
4. If the indicator digit is less than $5$ ($< 5$), keep the target digit unchanged and replace all digits to its right with zeros (or drop decimal tail digits).

Example: Round $6,749,825.438$ to the nearest ten-thousand.

• Target position: Ten-thousands place (digit is $4$, representing $40,000$).
• Indicator digit: Thousands place (digit is $9$).
• Since $9 \ge 5$, round up: $4$ becomes $5$. The result is $6,750,000$.

---

Integers & Signed Operations

The set of integers is defined as $\mathbb{Z} = {\dots, -4, -3, -2, -1, 0, 1, 2, 3, 4, \dots}$. It includes all positive whole numbers, zero, and negative whole numbers.

Visual Representation on a Horizontal Number Line

  Negative Direction (<--)                       (-->) Positive Direction
<---|-------|-------|-------|-------|-------|-------|-------|-------|--->
   -4      -3      -2      -1       0       1       2       3       4

• Absolute Value ($|x|$): The non-negative distance of a number from zero on the number line. For example, $|-7| = 7$ and $|7| = 7$.
• Opposites (Additive Inverses): Two numbers that are the same distance from zero but on opposite sides of the number line. Their sum is always zero: $a + (-a) = 0$.

Operational Rules for Integers

1. Addition & Subtraction

• Same Signs: Add their absolute values and retain the shared sign. $\text{Example: } (-6) + (-9) = -15$
• Different Signs: Subtract the smaller absolute value from the larger absolute value, and take the sign of the number with the larger absolute value. $\text{Example: } (-14) + 8 = -(14 - 8) = -6$
• Subtracting Integers: Subtraction is defined as adding the additive inverse. Change the subtraction sign to addition and invert the sign of the second term ($a - b = a + (-b)$). $\text{Example: } 7 - (-12) = 7 + (+12) = 19$ $\text{Example: } -8 - 5 = -8 + (-5) = -13$

2. Multiplication & Division

• Like Signs: The product or quotient of two numbers with the same sign is positive. $(+) \times (+) = + \quad \text{and} \quad (-) \times (-) = +$ $\text{Example: } (-8) \times (-6) = +48 \quad | \quad (-56) \div (-7) = +8$
• Unlike Signs: The product or quotient of two numbers with different signs is negative. $(+) \times (-) = - \quad \text{and} \quad (-) \times (+) = -$ $\text{Example: } (-9) \times 4 = -36 \quad | \quad 72 \div (-9) = -8$

---

Order of Operations: BEDMAS

When an arithmetic expression contains multiple operations, it must be evaluated following the formal order of operations codified by the acronym BEDMAS.

$\mathbf{B} \longrightarrow \mathbf{E} \longrightarrow \mathbf{D/M} \longrightarrow \mathbf{A/S}$

1. B — Brackets: Evaluate expressions inside grouping symbols first, working from the innermost brackets to the outermost brackets. Grouping symbols include parentheses $( )$, square brackets $[ ]$, curly braces ${ }$, and fraction bars.
2. E — Exponents: Evaluate all powers and exponential terms.
3. D and M — Division and Multiplication: Evaluate division and multiplication in order from left to right as they appear. Neither operation takes inherent priority over the other!
4. A and S — Addition and Subtraction: Evaluate addition and subtraction in order from left to right as they appear. Neither operation takes inherent priority over the other!

[!IMPORTANT] Critical MPT Rule on Left-to-Right Precedence
A widespread candidate mistake is evaluating multiplication before division regardless of order because 'M' comes after 'D' in BEDMAS, or vice versa. Division and Multiplication have equal precedence.
Consider $24 \div 4 \times 3$:

• Correct: Evaluate left to right $\rightarrow (24 \div 4) \times 3 = 6 \times 3 = 18$.
• Incorrect: Evaluating multiplication first $\rightarrow 24 \div (4 \times 3) = 24 \div 12 = 2$ (WRONG!).

---

High-Frequency MPT Traps & Pitfalls

Trap 1: Negative Bases vs. Negated Exponents

• Case A: $(-3)^2$ — The base is $(-3)$. Evaluated as $(-3) \times (-3) = +9$.
• Case B: $-3^2$ — The base is $3$. The negative sign acts as a coefficient of $-1$. Evaluated as $-(3 \times 3) = -9$.

Trap 2: Nested Brackets and Sign Distribution

When evaluating nested expressions such as $5 - 2[3 - (4 - 9)]$, candidate errors frequently stem from failing to distribute negative signs properly. Work strictly from the innermost parentheses outward.

---

Detailed Step-by-Step Worked Examples

Example 2.1.1: Multi-Step BEDMAS Evaluation with Nested Brackets

Evaluate the following integer expression completely: $-18 + 6 \times [(-3)^2 - (14 - 19)]$

Solution Step-by-Step:

1. Innermost Bracket: Evaluate $(14 - 19)$. $14 - 19 = -5$ Substitute back into the expression: $-18 + 6 \times [(-3)^2 - (-5)]$

2. Exponent Inside Bracket: Evaluate $(-3)^2$. $(-3)^2 = (-3) \times (-3) = 9$ Substitute back: $-18 + 6 \times [9 - (-5)]$

3. Outer Bracket Evaluation: Apply the double negative rule $9 - (-5) = 9 + 5 = 14$. $-18 + 6 \times [14]$

4. Multiplication: Evaluate $6 \times 14$. $6 \times 14 = 84$ Substitute back: $-18 + 84$

5. Addition: Combine the negative and positive integers. $-18 + 84 = +(84 - 18) = 66$

Final Answer: $66$

---

Example 2.1.2: Place Value & Multi-Tiered Rounding

Given the standard decimal number $7,429,851.364$, perform the following two operations:

1. Round the number to the nearest ten-thousand.
2. Round the original number to the nearest hundredth.
3. Find the difference between the rounded ten-thousands value and $7,400,000$.

Solution Step-by-Step:

1. Rounding to Nearest Ten-Thousand:

   • Target position: Ten-thousands digit is $2$ (representing $20,000$).
   • Indicator digit: Thousands digit is $9$.
   • Since $9 \ge 5$, round up: $2$ becomes $3$. Replace lower whole number digits with zeros and drop decimal terms.
   • Rounded value: $7,430,000$.

2. Rounding to Nearest Hundredth:

   • Target position: Hundredths digit is $6$ (representing $0.06$).
   • Indicator digit: Thousandths digit is $4$.
   • Since $4 < 5$, keep target digit $6$ unchanged and truncate.
   • Rounded value: $7,429,851.36$.

3. Calculating Difference: $7,430,000 - 7,400,000 = 30,000$

Final Answer: Ten-thousand rounded: $7,430,000$; Hundredth rounded: $7,429,851.36$; Difference: $30,000$.

---

Example 2.1.3: Complex Integer Expression with Left-to-Right Priority

Calculate the exact value of the following expression: $48 \div 4 \times (5 - 8)^2 - (-14 + 6)$

Solution Step-by-Step:

1. Brackets: Evaluate $(5 - 8)$ and $(-14 + 6)$. $5 - 8 = -3$ $-14 + 6 = -8$ Substitute back: $48 \div 4 \times (-3)^2 - (-8)$

2. Exponents: Evaluate $(-3)^2$. $(-3)^2 = 9$ Substitute back: $48 \div 4 \times 9 - (-8)$

3. Division & Multiplication (Left-to-Right):

   • First operation from left is division: $48 \div 4 = 12$.
   • Expression becomes: $12 \times 9 - (-8)$.
   • Next operation is multiplication: $12 \times 9 = 108$.
   • Expression becomes: $108 - (-8)$.

4. Subtraction / Addition: Apply double negative rule. $108 - (-8) = 108 + 8 = 116$

Final Answer: $116$